Graph of a function: Difference between revisions

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{{About||graph-theoretic representation of a function|Functional graph}}

{{more citations needed|date=August 2014}}

[[File:Polynomial of degree three.svg|class=skin-invert-image|thumb|250x250px|Graph of the function <math>f(x)=\frac{x^3+3x^2-6x-8}{4}.</math>]]

{{functions}}

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{{anchor|graph of a relation}}A graph of a function is a special case of a [[Relation (mathematics)|relation]].

In the modern [[foundations of mathematics]], and, typically, in [[set theory]], a function is actually equal to its graph.<ref name="Pinter2014">{{cite book|author=Charles C Pinter|title=A Book of Set Theory|url=https://books.google.com/books?id=iUT_AwAAQBAJ&pg=PA49|year=2014|orig-year=1971|publisher=Dover Publications|isbn=978-0-486-79549-2|pages=49}}</ref> However, it is often useful to see functions as [[Map (mathematics)|mappings]],<ref>{{cite book|author=T. M. Apostol|authorlink=Tom M. Apostol|title=Mathematical Analysis|year=1981|publisher=Addison-Wesley|page=35}}</ref> which consist not only of the relation between input and output, but also which set is the ___domain, and which set is the [[codomain]]. For example, to say that a function is onto ([[Surjective function|surjective]]) or not the codomain should be taken into account. The graph of a function on its own does not determine the codomain. It is common<ref>{{cite book|author=P. R. Halmos|title=A Hilbert Space Problem Book|url=https://archive.org/details/hilbertspaceprob00halm_811|url-access=limited|year=1982|publisher=Springer-Verlag|isbn=0-387-90685-1|page=[https://archive.org/details/hilbertspaceprob00halm_811/page/n47 31]}}</ref> to use both terms ''function'' and ''graph of a function'' since even if considered the same object, they indicate viewing it from a different perspective.

[[File:X^4 - 4^x.PNG|class=skin-invert-image|350px|thumb|Graph of the function <math>f(x) = x^4 - 4^x</math> over the [[Interval (mathematics)|interval]] [−2,+3]. Also shown are the two real roots and the local minimum that are in the interval.]]

== Definition ==

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=== Functions of two variables ===

[[File:F(x,y)=−((cosx)^2 + (cosy)^2)^2.PNG|class=skin-invert-image|thumb|250px|Plot of the graph of <math>f(x, y) = - \left(\cos\left(x^2\right) + \cos\left(y^2\right)\right)^2,</math> also showing its gradient projected on the bottom plane.]]

The graph of the [[trigonometric function]]

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Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function:

<math display=block>f(x, y) = -(\cos(x^2) + \cos(y^2))^2.</math>

== Generalizations ==

The graph of a function is contained in a [[Cartesian product]] of sets. An <math>X</math>–<math>Y</math> plane is a Cartesian product of two lines, called <math>X</math> and <math>Y,</math> while a cylinder is a cartesian product of a line and a circle, whose height, radius, and angle assign precise locations of the points. [[Fibre bundle]]s are not Cartesian products, but appear to be up close. There is a corresponding notion of a graph on a fibre bundle called a [[Section (fiber bundle)|section]].

The {{visible anchor|graph of a multifunction}}, say the [[Multivalued function|multifunction]] <math>\mathcal{R} : X \rightrightarrows Y,</math> is the set <math>\operatorname{gr} \mathcal{R} := \left\{ (x, y) \in X \times Y : y \in \mathcal{R}(x) \right\}.</math>

== See also ==